Tiêu đề 04- Triangles and its properties(1).pdf ✅

4 CHAPTER TRIANGLES AND ITS PROPERTIES CONTENTS • Triangle • Interior and Exterior of a Triangle • Types of Triangle • Angle sum property of a Triangle • Median & Altitude of a Triangle • Exterior Angle of a Triangle • Triangle Inequality • Pythagoras Theorem ➢ TRIANGLE A geometrical figure formed by joining three non- collinear points by three line segments is called a triangle. A B C The triangle ABC has : Sides : AB, BC , CA Vertices : A, B and C. Angles : BAC or CAB, ABC or CBA and ACB or BCA. A triangle is denoted by the symbol ''. The three sides and three angles taken together are called six elements or six parts of a triangle. ❖ EXAMPLES ❖ Ex.1 Do three collinear points A, B and C form a triangle ? Sol. No, three collinear points form a line. Ex.2 For the triangle LMN, name (a) the side opposite to M. (b) the angle opposite to side LM. (c) the vertex opposite to side NL. (d) the side opposite to vertex N. Sol. (a) The side opposite to M is LN. (b) The angle opposite to side LM is N. (c) The vertex opposite to side NL is M. (d) The side opposite to vertex N is LM. Ex.3 How many different triangles are in figure ? Name each of them. A B D F E C Sol. ABC,ADE,ABE,ADC,BFC,BFD, BDE,CEF,CED,DEF,BCD,BEC. So, there are 12 different triangles in the given figure.
➢ INTERIOR AND EXTERIOR OF A TRIANGLE Interior of a triangle is the region of the plane enclosed by ABC. Here, point P is in the interior of ABC. A B C P Q Exterior of a triangle is the region of the plane which lies beyond or not enclosed by the boundary of ABC. In figure, Q is the point which is in the exterior of the ABC. A B C P Q Triangular region Interior of ABC (as shown by the shaded region P in figure) together with the points on the boundary of ABC (as shown by point Q) is known as the triangular region ABC. ➢ TYPES OF TRIANGLE Based on angles Based on sides 1. Acute angled triangle : A triangle whose all angles are acute i.e., less than 90o. 60o 50o 70o A B C 1. Equilateral triangle : A triangle with all sides equal to one another. A B C 4 cm 4 cm 4 cm 2. Obtuse angled triangle : A triangle whose one angle is obtuse i.e., greater than 90o. 100o 30o 50o A D C A triangle cannot have more than one obtuse angle. 2. Isosceles triangle : A triangle with any two sides equal to each other. A B C 6 cm 5 cm 5 cm 3. Right angled triangle : A triangle whose one angle is of measure 90o also the other two angles are acute angles whose sum is 90o. 90o 30o 60o A B C The side of opposite to the right angle is called the hypotenuse, and other two side are called legs of the right triangle. 3. Scalene triangle : A triangle in which all sides are unequal. A B C 8 cm 6 cm 7 cm ➢ ANGLE SUM PROPERTY OF A TRIANGLE The sum of the angles of a triangle is 180o or two right angles. 1 2 3 5 4 P Q R S T Given : A triangle PQR. To prove : 1 + 2 + 3 = 180o i.e., sum of all angles of a triangle is 180o. Construction : Through P, draw a line ST parallel to QR. Proof : As ST || QR and transversal PQ cuts them.  2 = 5 (alternate angles) ...(1) Again ST || QR and transversal PR cuts them.  3 = 4 (alternate angles) ...(2) Adding (1) and (2), we get 2 + 3 = 5 + 4 ...(3) Now adding 1 on both sides to equation (3), we get 1 + 2 + 3 = 1 + 5 + 4  1 + 2 + 3 = 180o (as 1 + 5 + 4 = 180o)
Note : (i). Each angle of an equilateral triangle measures 60o (ii) The angles opposite to equal sides of an isosceles triangle are equal. (iii) A scalene triangle has all angles unequal. (iv) A triangle cannot have more than one right angle (v) A triangle cannot have more than one obtuse angle. (vi) In a right triangle, the sum of two acute angles is 90o. (vii)The sum of the lengths of the sides of a triangle is called perimeter of triangle. ❖ EXAMPLES ❖ Ex.4 Classify the triangles as Scalene, isosceles or equilateral, if their sides are : (i) 2 cm, 3 cm, 2 cm (ii) 2 cm, 2 cm, 2 cm (iii) 3 cm, 6 cm, 4 cm Sol. (i) As two sides are equal, so this is an isosceles triangle. (ii) As all sides are equal, so this is an equilateral triangle. (iii) As all sides are unequal, so this is a scalene triangle. Ex. 5 Classify the following triangles according to their angles : (i) 70o 30o 80o (ii) 130o 30o 20o (iii) 20o 90o 70o (iv) 40o 45o 95o (v) 40o 50o 90o (vi) 80o 75o 25o Sol. (i) As all the angles of this triangle are acute, so this is an acute triangle. (ii) As one of the angles (130°) is obtuse, so this is an obtuse triangle. (iii) As one of the angles is a right angle (90°), so this is a right triangle. (iv) As one of the angles is obtuse (95°), so this is an obtuse triangle. (v) As one of the angles is a right angle (90°), so this is a right triangle. (vi) As all the angles are acute, so this is an acute triangle. Ex. 6 Classify the triangles as acute, obtuse or right, whose angles are : (i) 50°, 40°, 90° (ii) 120°, 30°, 30° (iii) 70°, 60°, 50° Sol. (i) As one of the angles is a right angle, so this is a right triangle. (ii) As one of the angles is an obtuse angle, so this is an obtuse triangle. (iii) As all the angles are acute, so this is an acute triangle. Ex.7 Classify the triangles according to their given sides as scalene, isosceles or equilateral : (a) 3.5 cm, 4 cm, 4 cm (b) 6 cm, 7 cm, 9 cm (c) 6.2 cm, 6.2 cm, 6.2 cm Sol. (a) As two sides are equal so it is an isosceles triangle. (b) As all the sides are different so it is an scalene triangle. (c) As all the sides are equal so it is equilateral triangle. Ex.8 Classify the triangles as acute, obtuse or right if angles are : (a) 60o, 30o, 90o (b) 120o, 40o, 20o (c) 60o, 60o, 60o Sol. (a) As one angle of 90o so, it is a right triangle. (b) As one angle (120o) is greater than 90o i.e., obtuse, so it is an obtuse triangle. (c) As each angle is of 60o, so it is an equilateral triangle. Ex.9 Two angles of a triangle are of measures 70o and 30o. Find the measure of the third angle. Sol. Let PQR be a triangle such that P = 70o, Q = 30o. Then, we have to find the measure of third angle R.
As P + Q + R = 180o (angle sum property of triangle) 70o + 30o + R = 180o 100o + R = 180o R = 180o – 100o  R = 80o 30o 70o P Q R Ex.10 One of the angles of a triangle has measure 70o and the other two angles are equal. Find these two angles. Sol. Let PQR be a triangle such that : P = 70o and Q = R = x (let) As P + Q + R = 180o (angle sum property of ) 70o + x + x =180o 2x = 180o – 70o 2x = 110o x = 2 110o x = 55o P Q R 70o x x So, measure of each of remaining two angles is 55o. Ex.11 Write the (i) side opposite to the vertex Q of PQR (ii) angle opposite to the side LM of LMN (iii) vertex opposite to the side RT of RST. Sol. (i) The side opposite to vertex Q is PR. P Q R (ii) Angle opposite to side LM is N. L M N (iii) Vertex opposite to the side RT of RST is S. R S T Ex.12 In each of the following, the measures of three angles are given. State in which case the angles can possibly be those of a triangle : (i) 53°, 73°, 83° (ii) 59°, 12°, 109° (iii) 45°, 45°, 90° (iv) 30°, 120°, 30° Sol. (i) 53° + 73° + 83° = 209° > 180° Therefore, not possible (ii) 59° + 12° + 109° = 180° Therefore, possible (iii) 45° + 45° + 90° = 180° Therefore, possible (iv) 30° + 120° + 30° = 180° Therefore, possible Ex.13 The three angles of a triangle are equal to one another. What is the measure of each angle ? Sol. Let each angle be of measure x in degrees. Then, by angle sum property x + x + x = 180°  3x = 180°  x = 60° So, the measure of each angle is 60°.